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a) Find a $2 \times 2$ real matrix $A$ that has an eigenvalue $\lambda_{1}=1$ with eigenvector $E_{1}=$ $\left(\begin{array}{l}1 \\ 2\end{array}\right)$ and an eigenvalue $\lambda_{2}=-1$ with eigenvector $E_{2}=\left(\begin{array}{l}2 \\ 1\end{array}\right)$.

b) Compute the determinant of $A^{10}+A$.
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